Chebyshev Polynomials

Chebyshev Polynomials: \( T_n(x) \)

Chebyshev polynomials are very useful for interpolating functions. Formally, the Chebyshev polynomial of degree \(n\) is defined as

\begin{equation*}
T_n(x) = \cos(n\cos^{-1}x), \qquad \text{for } x\in [-1,\;1]
\end{equation*}
At first look, this expression does not resemble a polynomial at all!
In this note we will follow two different approaches to show that \(T_n(x)\)is indeed a polynomial. We start with the easy one, which only requires some basic trigonometric identities in real numbers, and where we will find a recursive definition of the Chebyshev polynomials. The not-so-easy approach requires working with complex numbers, but it will give us a closed formula for computing the polynomials.

On the other hand, the zeros of the polynomial occur when \(n\cos^{-1}x = k\pi + \frac{\pi}{2}\)for some integer \(k\):
\begin{equation*}
x = \cos\left(\frac{2k-1}{2n}\pi\right),\qquad k=1,\dots,n
\end{equation*}